- Email: [email protected]
Characterization of the mode I fracture energy of adhesive joints
ARTICLE IN PRESS
International Journal of Adhesion & Adhesives 26 (2006) 644–650 www.elsevier.com/locate/ijadhadh
Characterization of the mode I fracture energy of adhesive joints G. Steinbrechera,, A. Buchmana, A. Sidessa, D. Shermanb a
RAFAEL Ltd., Armament Development Authority Ltd., P.O.B. 2250, Haifa, Israel Department of Materials Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
b
Accepted 26 September 2005 Available online 3 February 2006
Abstract This investigation was aimed at improving the calculation of the Mode I fracture energy, GIC, of adhesive joints by incorporating the elasticity of the adhesive layer. It was also aimed at proposing ways to improve the calculation of GIC over the existing standard for the measurement of that material property. Our experiments were performed with a variety of aluminum specimens bonded with a large number of different adhesives in various thicknesses. All specimens were tested under mode I loading. In calculating GIC, it is common to neglect the properties of the adhesive layer, especially when the compliance of the system is considered. The validity of this attitude was tested in the present study, and it was found to be in accord with the experimental results but only for joints bonded with a relatively thin layer of adhesive. A method for improving the calculation of the fracture energy of standard specimens, using a theoretical model for the compliance is proposed. This method comprises all the adhesive parameters and is appropriate for linear elastic joints. r 2006 Elsevier Ltd. All rights reserved. Keywords: Adhesives joints; Fracture energy; Double cantilever beam specimen
1. Introduction The use of adhesive joints in load-bearing components has been growing steadily in recent years. The said practice bears some major advantages over other methods. Among these are the possibility to join dissimilar materials and the ability to absorb residual strains created as a result of thermal stresses, their high strength-to-weight ratio, low cost, etc. As the use of adhesive joints in structural parts becomes more frequent, the need to define rigorous failure criteria grows more crucial. There are two basic disciplines for categorizing failure criteria in adhesive joints. The first and more common one is based on the characterization of the stresses acting on the bonded pieces and on a definition of the maximum force that can be applied to a joint in each of the four major types of loading: tension, shear, peel, and cleavage. However, although this discipline is relatively easy to apply, it is applicable mostly to simple geometries. MoreCorresponding author. Tel.: +972 4 8795003; fax: +972 4 8794782.
E-mail address: [email protected] (G. Steinbrecher). 0143-7496/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2005.09.006
over, it is not capable of dealing with defects that are commonly present in adhesive joints. The other discipline, which is the focus of the present investigation, is based on fracture mechanics theory, the central argument of which specifies that the maximum stress, or load, that can be applied to a material is determined by the defects it contains. The relevant failure criteria are either the fracture toughness (KC) or the critical strain energy release rate (GC). The latter describes the energy required to initiate a crack from an existing defect. The purpose of the present research was to characterize the fracture energy of adhesive joints and in particular to check the validity of the assumptions that are implied by the existing standard for measuring GC. Many methods have been described for measuring the strain energy release rate (GC) of adhesive joints, being both analytical and numerical solutions for that purpose, whether under mode I loading, e.g. [1,2], or under mixed mode loading [3–5]. Consequently, many types of specimens have been developed in order to measure GC of adhesive joints, the most common being the double cantilever beam (DCB) [6–9]. The main reasons for its
ARTICLE IN PRESS G. Steinbrecher et al. / International Journal of Adhesion & Adhesives 26 (2006) 644–650
wide usage are its simplicity both of production and of analysis, and the possibility of easily controlling the different parameters of the specimen, e.g. the thickness of the adhesive. In addition, a small variation in loading can be applied on the same specimen to achieve mixed-mode conditions. An essential requirement for the standard DCB specimen is the linear elastic deformation of the system, in order that linear-elastic fracture mechanics (LEFM) analysis can be applied. In this case, the critical strain energy release rate is calculated with GC ¼
P2C dC , 2B da
(1)
where PC is the critical load for crack initiation, C the specimen’s compliance, B the width of the specimen, and a the crack length. When evaluating the derivative of the compliance, dC/da, it is usually assumed that the contribution of the adhesive layer to the overall compliance is negligible. Based on this assumption, a common expression for dC/da is 2 dC 8 3a 1 ¼ þ , da E S B h3 h where ES is the elastic modulus of the beam and h half the thickness of the bonded DCB specimen. The first term in the parenthesis originates from the deflection of a cantilever beam; the second corresponds to a finite element correction [10] due to the shear stresses existing when the beam length-to-height ratio is finite. As can be seen, the expression for dC/da does not include any parameter of the adhesive layer, which means that a specimen’s compliance does not depend neither on the type of adhesive used nor on the thickness of the adhesive layer. Applying the expression for dC/da to (1), the commonly used expression for GIC can now be deduced as G IC ¼
4P2C ð3a2 þ h2 Þ . E S B2 h3
(2)
Eq. (2) is the one used in ASTM standard D-3433 for evaluating GIC of adhesive joints [11]. The usual practice is to conduct several tests in which the PC values are measured for different crack lengths and to calculate GIC using Eq. (2). Since the crack propagation in that kind of test is usually stable, several tests can be made using the same sample. GIC calculated in accordance with ASTM standard D-3433 seems to vary with the crack length, see [12] for aluminum/epoxy DCB joints. However, since GIC is a material property, such apparent dependence is unacceptable. Several of the explanations advanced for that dependence of GIC in composite materials, e.g. [13], are related to a bridging mechanism, but we consider such a mechanism to be irrelevant where bonded metal joints are concerned.
645
A common approach to evaluating GIC is to use the compliance calibration method, the main premise of which is that in reality there are many factors that can affect the specimen’s behavior and that it is therefore plausible to measure rather than to calculate the compliance function. In practice specimens are fabricated with a variety of crack lengths, and their compliance is measured experimentally. A plot of the compliance (usually C1/3) versus the crack length produces the empirical dependence of the compliance on the crack length, see [14–16] to list a few. The most important drawback of that empirical procedure is that the parameters effecting the deviation from beam theory are not well understood. Thus, any change in the specimen’s parameters (e.g. thickness, type of adhesive, etc.) will require a new series of tests in order to validate the effect of the specimen’s compliance. A comprehensive study that addresses the implications associated with GIC measurement has recently been published [17]. This investigation was aimed at proposing a more accurate expression to describe the compliance of a DCB specimen, to check the validity of the assumptions used in the ASTM standard, and, as a result, to establish a method for a more accurate evaluation of the GIC of adhesive joints. An expression for the compliance of a DCB specimen that takes into account the adhesive parameters was proposed by Penado [1]. The specimen is modelled as a beam partially supported by an elastic foundation. The corrected expression is ( 2 8 a3 3 h 1 E1 h C¼ 1þ þ þ3 2 0:25 a 0:5 ~ E1B h a 8G Kh 1 K~ h ) 3 3 h þ 3 , ð3Þ 0:75 a 2K~ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 where K~ ¼ 4 3k=E 1 B, k ¼ ½k1 adherend þ k adhesive , kadherend ¼ 2 4E 1 B=h and kadhesive ¼ B=tE 2 =ð1 n2 Þ. The notations in Eq. (3) refer to the adherend when the subscript ‘‘1’’ is used and to the adhesive when the subscript is ‘‘2’’. E is Young’s modulus, G the shear modulus, and n Poisson’s ratio. Note that h is the thickness of the adherend, t that of the adhesive. Eq. (3), as distinct from the one used by ASTM [11], is taking the parameters of the adhesive into account. In [1] the model is verified using finite element analysis. In the present work, the model is verified experimentally and is used for an assessment of GIC. 2. Experimental Double cantilever beam specimens, schematically shown in Fig. 1, were fabricated by bonding 254 mm 25.4 mm 4.9 mm aluminum 6061—T3 beams. The adherends were pretreated with chromatic anodization without sealing. A thin polyimide film was attached to one end of the aluminium beam prior to the bonding, to serve as a ‘starter
ARTICLE IN PRESS 646
G. Steinbrecher et al. / International Journal of Adhesion & Adhesives 26 (2006) 644–650
Uralite 3150 (Hexcel, USA)—a two-component rubberlike polyurethane adhesive (E ¼ 0:13 GPa).
3. Results
Fig. 1. Double cantilever beam (DCB) specimen.
notch’. Copper wires of the appropriate diameter were placed on the opposite ends of the adherend in order to control the bond thickness. The specimens were given a short precrack by driving a chisel between the adherends. Each data set included 3–5 identical specimens and each one was used for a number of crack extension events, since the crack propagates in a stable manner. The initial crack propagation, started at the chisel pre crack, was not used for the G1C evaluations. The specimens were bonded individually and the side panels of each specimen were polished after curing and then painted using a white correction fluid in order to enable a clear view of the crack tip. During the tests, the crack tip was identified using a magnifying glass and the crack length was measured between the end of the polyimide film and the crack tip. Prior to the loading, end blocks were glued to the specimen as shown in Fig. 1. The loads were applied through drill holes in the blocks. The correction factors [17] for the influence of the end-blocks (N) and for large displacements (F) were not taken into account in the G1C calculations, since their ratio, F/N was found to be close to 1 (F =N ¼ 0:98 for short normalized crack lengths of a/L0.2). Several different adhesives were used for the DCB tests. The elastic modulus of each adhesive was measured using 5 tensile bulk ASTM D-638 specimens. Given are the average values for the tensile modulus. In general, all results fell within a boundary of 75% around the average value.
Helmipur T700/500LF (Helmitin, Germany)—a twocomponent polyurethane adhesive, containing 62% CaCO3 filler (E ¼ 0:4 GPa). Scotch Weld 3549 (3 M)—a rubber-like, two-component polyurethane adhesive (E ¼ 0:017 GPa). Hysol 4183/3561 (Loctite, formerly Dexter Corp.)—a tough two-component casting system (E ¼ 6:8 GPa). EN-2 (Conap, USA)—a rubber-like two-component polyurethane adhesive (E ¼ 0:0046 GPa). PlioGrip7770 (Ashland Chemicals, USA)—a two-component polyurethane adhesive (E ¼ 0:707 GPa). FM-73 (American Cyanamid, USA)—a tough structural adhesive (E ¼ 1:6 GPa).
Two typical examples of the calculation of GIC in accordance with the ASTM standard are shown in Fig. 2. The results refer to aluminum DCB joints bonded with two polyurethane adhesives. The bond thickness was 0.11 mm. GIC was calculated using Eq. (2), i.e. by the ASTM method. In this test and in all others the crack propagated in the adhesive layer unless otherwise stated. The apparent dependence of GIC on the crack length is clearly seen with respect to both specimens. Furthermore, wide scatter of the measured GIC is evident. To characterize the difference between the actual compliance of the DCB specimens and that predicted by beam theory, several tests were carried out in which the specimens were loaded within their elastic regime. Fig. 3 depicts the measured values of the compliance as compared with those predicted by beam theory for DCBs with two different adhesives and several bond layer thicknesses. Since beam theory neglects the adhesive’s parameters, the predicted compliance (solid lines) is nearly the same for all specimens. However, it is evident that the measured compliances are higher than the ones predicted and that the differences are proportional to the bond layer thickness. Similar experimental results of compliance vs. crack length for three different adhesives are plotted in Fig. 4, but this time for compliances calculated using Eq. (3), which was derived from Penado’s model [1]. That model considers the properties of both adhesive and adherends. A much better agreement between the experimental results and the
Fig. 2. GIC vs. crack length calculated in accordance with ASTM standard for DCB aluminum specimens with two different adhesives.
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were selected: (i) beam theory where CðaÞ ¼ 8a3 =EBh3 , (ii) the short-beam correction, CðaÞ ¼ 8a3 =EBh3 þ 8a=EBh, and (iii) beam on elastic foundation, Eq. (3). Figs. 5(a) and (b) reveal the calculated compliance vs. crack length for two extreme cases: a DCB joint bonded with a stiff adhesive (Hysol 4183/3561) and the same joint, but bonded with a rubber-like adhesive (EN-2). It is evident that the contribution of the short beam correction used by the ASTM standard [11] is negligible for the selected geometry. On the other hand, neglecting the adhesive’s contribution to the compliance can result in a large discrepancy (up to 100% for short cracks and
Fig. 3. The measured compliance compared with those predicted for aluminum DCB specimen bonded with different adhesives of various thicknesses.
Fig. 4. Measured compliance compared with that predicted by the model, Eq. (3), for aluminum DCB specimens bonded with two different adhesives of various thicknesses of the adhesive layer.
model’s predictions is readily seen, confirming that the use of an adequate analytical model for determining the compliance can save the time required for the lengthy compliance calibration procedure every time a change is made in the specimen. In the course of the present work, tests were carried out using 10 different adhesives. In order to ascertain the differences between the various existing techniques for the compliance function calculations, three different methods
Fig. 5. The compliance function of a DCB specimen according to simple beam theory, the correction for a short beam, and the beam on elastic foundation model for a tough adhesive (a), and a ductile adhesive (b).
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rubber-like adhesives) between the beam-theory-based prediction and the actual results. Since GIC is proportional to the derivative of the compliance function, dC/da, examination of the effect of the method of calculation on dC/da is essential. Figs. 6(a) and (b) represent the values of dC/da as calculated, analytically, using each of the three methods listed above. It is readily seen that with short cracks in a ductile adhesive, the differences can reach up to 75%. According to Eq. (1), GIC depends on two parameters: the derivative of the compliance function (dC/da), and the threshold loads for crack initiation, PC. As stated above, the first can be better calculated using the beam-on-elasticfoundation model. PC is obtained experimentally. We further propose a procedure for the improved derivation of GIC from the experimental results: According to fracture mechanics theory, the GIC vs. load function, Eq. (1), can be
presented as follows: 1 P2C dC ¼ GIC . da 2B Thus, plotting P2C =2B for different crack lengths against (dC/da)1 results in a linear plot that should intersect the origin, provided the joint’s behaviour is indeed linearelastic. The slope of this line represents the value of GIC. The graph of P2C =2B values obtained experimentally vs. (dC/da)1 calculated with Penado’s model, Eq. (3), for a DCB specimen made of aluminum/PG7770 is shown in Fig. 7(a). The same experimental results were used to obtain GIC values in accordance with the ASTM-D-3433 method, the result being shown in Fig. 7(b). It is clearly seen that the resulting line in Fig. 7(a) is linear and intersects the origin, which means that the behaviour of the system is indeed linear-elastic. The slope of this line is the GIC, and its value is also shown in Fig. 7(a). The average value of GIC as obtained using the ASTM standard, Fig. 7(b), is significantly lower than the one calculated with Eq. (3). Two other examples, each for a different system, are given in Figs. 7(c)–(f). The calculated GIC by the current method and by the ASTM standard of 7 additional adhesive joints are shown in Table 1. All the data show that GIC calculation based on ASTM standard underestimates the fracture energy, with much larger scatter. 4. Discussion
Fig. 6. The derivative of compliance vs. crack length for a DCB specimen using different models for a tough adhesive (a), and a ductile adhesive (b).
The existing standard [11] for measuring the fracture energy, GIC, of adhesive joints is based on linear elastic fracture mechanics and implicitly assumes that the adhesive layer’s parameters are negligible when calculating the compliance of the DCB specimen. The validity of this assumption was examined in the present investigation, and it was found to conflict with the experimental results obtained, specifically for joints that have a relatively thick adhesive layer. The compliance calibration method, which is frequently used, has two major disadvantages; the need for a large number of specimens and the requirement to reassess the compliance every time even a minor change is made in the specimens. Furthermore, the empirical results obtained include both the adherend’s and the adhesive’s contributions to the overall compliance, and it is impossible to distinguish between the two. The methods suggested in the recent round-robin [17] seem to overcome some of the problems existing in the ASTM standard, but do not resolve the need for an experimental evaluation of the compliance function. A method for calculating GIC is proposed in the present investigation. This method is based on combining the experimentally obtained measurements of the critical load for crack initiation with an analytical model to calculate the compliance of the specimens more accurately. It was found that the values obtained for GIC in our specimens were always higher than the GIC values obtained using the
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Fig. 7. GIC calculated in accordance with the proposed technique compared with that determined with the ASTM method for the following systems: aluminum/PG7770 DCB specimen—(a) and (b), aluminum/FM-73 DCB—(c) and (d), and aluminum/U-3150 DCB—(e) and (f).
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Table 1 GIC values of several adhesive joints as calculated using the suggested method and by ASTM D-3433 Adhesive used
GIC (suggested method) (J/m2)
GIC (ASTM D-3433) (J/m2)
Helmipur 700T/LF500 Hysol 4183/3561 E-433 Uralite 3125 Epon 828/V140 Fusor 305 EN-2
980770 682730 615740 260720 206710 188713 181720
8877179 457771 4127158 178746 195726 178770 125740
ASTM standard, due to the adhesive parameters in the latter being neglected. In fact, this result could be predicted from Figs. 6(a) and (b): the measured compliance and its derivative were always higher than those resulting from beam theory as used by the ASTM standard. Since GIC is proportional to the derivative of the compliance function, the resulting GIC will also be higher. It is evident that the ASTM method may be inadequate for measuring the fracture energy of adhesive joints with a thick and relatively compliant bond layer. Describing the DCB specimen as a beam on elastic foundation was found to give an accurate estimate for the compliance of DCB specimens. Finally, in our method the fracture energy was found to be independent of the crack length. It is noted that the elastic properties of the adhesives used are required for our calculations. Furthermore, our method applies only to adhesive joints that are within the LEFM framework. The possibilities to incorporate this method for non-linear adhesives or adhesives with R-curve behavior need further examination. 5. Summary The common practice for measuring GIC of adhesive joints, as described by the ASTM standard, has several
disadvantages. The typical results show a large diversity and can, moreover, lead to different values of GIC for different crack lengths. Such results are liable to preclude the use of GIC as a failure criterion and thus give rise to the use of the empirical compliance calibration technique, which is not very effective when testing many parameters of the adhesive joint. The approach used in this work introduces the use of a known analytical model for the improved calculation of the system’s compliance by including the adhesive’s parameters. References [1] Penado E. J Compos Mat 1993;27(4):383–407. [2] Liu Z, Gibson RF, Newaz GM. In: Proceedings of 31st International SAMPE technical conference, Society for the Advancement of Material and Process Engineering, Chicago, IL, October 1999. [3] Williams JG. Int J Fracture 1988;36:101–19. [4] Krenk S. Eng Fracture Mech 1992;43(4):549–59. [5] Fernlund G, Spelt JK. Composites Sci Tech 1994;50:441–9. [6] Kinloch AJ, Little MSG, Watts JF. Acta Mater 2000;48:4543–53. [7] Blackman BRK, Kinloch AJ, Taylor AC, Wang Y. J Mater Sci 2000;35(8):1867–84. [8] Curley AJ, Hadavinia H, Kinloch AJ, Taylor AC. Int J Fracture 2000;103:41–69. [9] Jacobsen TK, Sorensen BF. Composites: Part A 2001;32:1–11. [10] Trantina C. J Compos Mater 1972;6:371. [11] ‘‘Fracture strength in cleavage of adhesives in bonded metal joints’’, ASTM-D3433-99, Annual Book of ASTM Standards (American Society for Testing and Materials, Easton, MD). [12] Liechti KM, Freda T. J Adhesion 1989;28:145–69. [13] Park BY, Kim SC, Jung B. Polymers Adv Technol 1996;8:371–7. [14] Rakestraw RD, Vrana MA, Dillard DA, Ward TC, Dillard JG. A fracture mechanics methodology for the testing and prediction of adhesive bond durability, AD-Vol 43, Durability and Damage Tolerance, ASME 1994. [15] Butkus LM. Environmental Durability of Adhesively Bonded Joints, Ph.D. Thesis, Georgia Institute of Technology, 1997. [16] Dalmas D, Laksimi L. Appl Compos Mater 1999;6:327–40. [17] Blackman BRK, Kinloch AJ, Paraschi M, Teo WS. Measuring the Mode I adhesive fracture energy, GIC, of structural adhesive joints: the results of an international round-robin. Int J Adhesion Adhesives 2003;23:293–305.
International Journal of Adhesion & Adhesives 26 (2006) 644–650 www.elsevier.com/locate/ijadhadh
Characterization of the mode I fracture energy of adhesive joints G. Steinbrechera,, A. Buchmana, A. Sidessa, D. Shermanb a
RAFAEL Ltd., Armament Development Authority Ltd., P.O.B. 2250, Haifa, Israel Department of Materials Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
b
Accepted 26 September 2005 Available online 3 February 2006
Abstract This investigation was aimed at improving the calculation of the Mode I fracture energy, GIC, of adhesive joints by incorporating the elasticity of the adhesive layer. It was also aimed at proposing ways to improve the calculation of GIC over the existing standard for the measurement of that material property. Our experiments were performed with a variety of aluminum specimens bonded with a large number of different adhesives in various thicknesses. All specimens were tested under mode I loading. In calculating GIC, it is common to neglect the properties of the adhesive layer, especially when the compliance of the system is considered. The validity of this attitude was tested in the present study, and it was found to be in accord with the experimental results but only for joints bonded with a relatively thin layer of adhesive. A method for improving the calculation of the fracture energy of standard specimens, using a theoretical model for the compliance is proposed. This method comprises all the adhesive parameters and is appropriate for linear elastic joints. r 2006 Elsevier Ltd. All rights reserved. Keywords: Adhesives joints; Fracture energy; Double cantilever beam specimen
1. Introduction The use of adhesive joints in load-bearing components has been growing steadily in recent years. The said practice bears some major advantages over other methods. Among these are the possibility to join dissimilar materials and the ability to absorb residual strains created as a result of thermal stresses, their high strength-to-weight ratio, low cost, etc. As the use of adhesive joints in structural parts becomes more frequent, the need to define rigorous failure criteria grows more crucial. There are two basic disciplines for categorizing failure criteria in adhesive joints. The first and more common one is based on the characterization of the stresses acting on the bonded pieces and on a definition of the maximum force that can be applied to a joint in each of the four major types of loading: tension, shear, peel, and cleavage. However, although this discipline is relatively easy to apply, it is applicable mostly to simple geometries. MoreCorresponding author. Tel.: +972 4 8795003; fax: +972 4 8794782.
E-mail address: [email protected] (G. Steinbrecher). 0143-7496/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijadhadh.2005.09.006
over, it is not capable of dealing with defects that are commonly present in adhesive joints. The other discipline, which is the focus of the present investigation, is based on fracture mechanics theory, the central argument of which specifies that the maximum stress, or load, that can be applied to a material is determined by the defects it contains. The relevant failure criteria are either the fracture toughness (KC) or the critical strain energy release rate (GC). The latter describes the energy required to initiate a crack from an existing defect. The purpose of the present research was to characterize the fracture energy of adhesive joints and in particular to check the validity of the assumptions that are implied by the existing standard for measuring GC. Many methods have been described for measuring the strain energy release rate (GC) of adhesive joints, being both analytical and numerical solutions for that purpose, whether under mode I loading, e.g. [1,2], or under mixed mode loading [3–5]. Consequently, many types of specimens have been developed in order to measure GC of adhesive joints, the most common being the double cantilever beam (DCB) [6–9]. The main reasons for its
ARTICLE IN PRESS G. Steinbrecher et al. / International Journal of Adhesion & Adhesives 26 (2006) 644–650
wide usage are its simplicity both of production and of analysis, and the possibility of easily controlling the different parameters of the specimen, e.g. the thickness of the adhesive. In addition, a small variation in loading can be applied on the same specimen to achieve mixed-mode conditions. An essential requirement for the standard DCB specimen is the linear elastic deformation of the system, in order that linear-elastic fracture mechanics (LEFM) analysis can be applied. In this case, the critical strain energy release rate is calculated with GC ¼
P2C dC , 2B da
(1)
where PC is the critical load for crack initiation, C the specimen’s compliance, B the width of the specimen, and a the crack length. When evaluating the derivative of the compliance, dC/da, it is usually assumed that the contribution of the adhesive layer to the overall compliance is negligible. Based on this assumption, a common expression for dC/da is 2 dC 8 3a 1 ¼ þ , da E S B h3 h where ES is the elastic modulus of the beam and h half the thickness of the bonded DCB specimen. The first term in the parenthesis originates from the deflection of a cantilever beam; the second corresponds to a finite element correction [10] due to the shear stresses existing when the beam length-to-height ratio is finite. As can be seen, the expression for dC/da does not include any parameter of the adhesive layer, which means that a specimen’s compliance does not depend neither on the type of adhesive used nor on the thickness of the adhesive layer. Applying the expression for dC/da to (1), the commonly used expression for GIC can now be deduced as G IC ¼
4P2C ð3a2 þ h2 Þ . E S B2 h3
(2)
Eq. (2) is the one used in ASTM standard D-3433 for evaluating GIC of adhesive joints [11]. The usual practice is to conduct several tests in which the PC values are measured for different crack lengths and to calculate GIC using Eq. (2). Since the crack propagation in that kind of test is usually stable, several tests can be made using the same sample. GIC calculated in accordance with ASTM standard D-3433 seems to vary with the crack length, see [12] for aluminum/epoxy DCB joints. However, since GIC is a material property, such apparent dependence is unacceptable. Several of the explanations advanced for that dependence of GIC in composite materials, e.g. [13], are related to a bridging mechanism, but we consider such a mechanism to be irrelevant where bonded metal joints are concerned.
645
A common approach to evaluating GIC is to use the compliance calibration method, the main premise of which is that in reality there are many factors that can affect the specimen’s behavior and that it is therefore plausible to measure rather than to calculate the compliance function. In practice specimens are fabricated with a variety of crack lengths, and their compliance is measured experimentally. A plot of the compliance (usually C1/3) versus the crack length produces the empirical dependence of the compliance on the crack length, see [14–16] to list a few. The most important drawback of that empirical procedure is that the parameters effecting the deviation from beam theory are not well understood. Thus, any change in the specimen’s parameters (e.g. thickness, type of adhesive, etc.) will require a new series of tests in order to validate the effect of the specimen’s compliance. A comprehensive study that addresses the implications associated with GIC measurement has recently been published [17]. This investigation was aimed at proposing a more accurate expression to describe the compliance of a DCB specimen, to check the validity of the assumptions used in the ASTM standard, and, as a result, to establish a method for a more accurate evaluation of the GIC of adhesive joints. An expression for the compliance of a DCB specimen that takes into account the adhesive parameters was proposed by Penado [1]. The specimen is modelled as a beam partially supported by an elastic foundation. The corrected expression is ( 2 8 a3 3 h 1 E1 h C¼ 1þ þ þ3 2 0:25 a 0:5 ~ E1B h a 8G Kh 1 K~ h ) 3 3 h þ 3 , ð3Þ 0:75 a 2K~ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 where K~ ¼ 4 3k=E 1 B, k ¼ ½k1 adherend þ k adhesive , kadherend ¼ 2 4E 1 B=h and kadhesive ¼ B=tE 2 =ð1 n2 Þ. The notations in Eq. (3) refer to the adherend when the subscript ‘‘1’’ is used and to the adhesive when the subscript is ‘‘2’’. E is Young’s modulus, G the shear modulus, and n Poisson’s ratio. Note that h is the thickness of the adherend, t that of the adhesive. Eq. (3), as distinct from the one used by ASTM [11], is taking the parameters of the adhesive into account. In [1] the model is verified using finite element analysis. In the present work, the model is verified experimentally and is used for an assessment of GIC. 2. Experimental Double cantilever beam specimens, schematically shown in Fig. 1, were fabricated by bonding 254 mm 25.4 mm 4.9 mm aluminum 6061—T3 beams. The adherends were pretreated with chromatic anodization without sealing. A thin polyimide film was attached to one end of the aluminium beam prior to the bonding, to serve as a ‘starter
ARTICLE IN PRESS 646
G. Steinbrecher et al. / International Journal of Adhesion & Adhesives 26 (2006) 644–650
Uralite 3150 (Hexcel, USA)—a two-component rubberlike polyurethane adhesive (E ¼ 0:13 GPa).
3. Results
Fig. 1. Double cantilever beam (DCB) specimen.
notch’. Copper wires of the appropriate diameter were placed on the opposite ends of the adherend in order to control the bond thickness. The specimens were given a short precrack by driving a chisel between the adherends. Each data set included 3–5 identical specimens and each one was used for a number of crack extension events, since the crack propagates in a stable manner. The initial crack propagation, started at the chisel pre crack, was not used for the G1C evaluations. The specimens were bonded individually and the side panels of each specimen were polished after curing and then painted using a white correction fluid in order to enable a clear view of the crack tip. During the tests, the crack tip was identified using a magnifying glass and the crack length was measured between the end of the polyimide film and the crack tip. Prior to the loading, end blocks were glued to the specimen as shown in Fig. 1. The loads were applied through drill holes in the blocks. The correction factors [17] for the influence of the end-blocks (N) and for large displacements (F) were not taken into account in the G1C calculations, since their ratio, F/N was found to be close to 1 (F =N ¼ 0:98 for short normalized crack lengths of a/L0.2). Several different adhesives were used for the DCB tests. The elastic modulus of each adhesive was measured using 5 tensile bulk ASTM D-638 specimens. Given are the average values for the tensile modulus. In general, all results fell within a boundary of 75% around the average value.
Helmipur T700/500LF (Helmitin, Germany)—a twocomponent polyurethane adhesive, containing 62% CaCO3 filler (E ¼ 0:4 GPa). Scotch Weld 3549 (3 M)—a rubber-like, two-component polyurethane adhesive (E ¼ 0:017 GPa). Hysol 4183/3561 (Loctite, formerly Dexter Corp.)—a tough two-component casting system (E ¼ 6:8 GPa). EN-2 (Conap, USA)—a rubber-like two-component polyurethane adhesive (E ¼ 0:0046 GPa). PlioGrip7770 (Ashland Chemicals, USA)—a two-component polyurethane adhesive (E ¼ 0:707 GPa). FM-73 (American Cyanamid, USA)—a tough structural adhesive (E ¼ 1:6 GPa).
Two typical examples of the calculation of GIC in accordance with the ASTM standard are shown in Fig. 2. The results refer to aluminum DCB joints bonded with two polyurethane adhesives. The bond thickness was 0.11 mm. GIC was calculated using Eq. (2), i.e. by the ASTM method. In this test and in all others the crack propagated in the adhesive layer unless otherwise stated. The apparent dependence of GIC on the crack length is clearly seen with respect to both specimens. Furthermore, wide scatter of the measured GIC is evident. To characterize the difference between the actual compliance of the DCB specimens and that predicted by beam theory, several tests were carried out in which the specimens were loaded within their elastic regime. Fig. 3 depicts the measured values of the compliance as compared with those predicted by beam theory for DCBs with two different adhesives and several bond layer thicknesses. Since beam theory neglects the adhesive’s parameters, the predicted compliance (solid lines) is nearly the same for all specimens. However, it is evident that the measured compliances are higher than the ones predicted and that the differences are proportional to the bond layer thickness. Similar experimental results of compliance vs. crack length for three different adhesives are plotted in Fig. 4, but this time for compliances calculated using Eq. (3), which was derived from Penado’s model [1]. That model considers the properties of both adhesive and adherends. A much better agreement between the experimental results and the
Fig. 2. GIC vs. crack length calculated in accordance with ASTM standard for DCB aluminum specimens with two different adhesives.
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were selected: (i) beam theory where CðaÞ ¼ 8a3 =EBh3 , (ii) the short-beam correction, CðaÞ ¼ 8a3 =EBh3 þ 8a=EBh, and (iii) beam on elastic foundation, Eq. (3). Figs. 5(a) and (b) reveal the calculated compliance vs. crack length for two extreme cases: a DCB joint bonded with a stiff adhesive (Hysol 4183/3561) and the same joint, but bonded with a rubber-like adhesive (EN-2). It is evident that the contribution of the short beam correction used by the ASTM standard [11] is negligible for the selected geometry. On the other hand, neglecting the adhesive’s contribution to the compliance can result in a large discrepancy (up to 100% for short cracks and
Fig. 3. The measured compliance compared with those predicted for aluminum DCB specimen bonded with different adhesives of various thicknesses.
Fig. 4. Measured compliance compared with that predicted by the model, Eq. (3), for aluminum DCB specimens bonded with two different adhesives of various thicknesses of the adhesive layer.
model’s predictions is readily seen, confirming that the use of an adequate analytical model for determining the compliance can save the time required for the lengthy compliance calibration procedure every time a change is made in the specimen. In the course of the present work, tests were carried out using 10 different adhesives. In order to ascertain the differences between the various existing techniques for the compliance function calculations, three different methods
Fig. 5. The compliance function of a DCB specimen according to simple beam theory, the correction for a short beam, and the beam on elastic foundation model for a tough adhesive (a), and a ductile adhesive (b).
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rubber-like adhesives) between the beam-theory-based prediction and the actual results. Since GIC is proportional to the derivative of the compliance function, dC/da, examination of the effect of the method of calculation on dC/da is essential. Figs. 6(a) and (b) represent the values of dC/da as calculated, analytically, using each of the three methods listed above. It is readily seen that with short cracks in a ductile adhesive, the differences can reach up to 75%. According to Eq. (1), GIC depends on two parameters: the derivative of the compliance function (dC/da), and the threshold loads for crack initiation, PC. As stated above, the first can be better calculated using the beam-on-elasticfoundation model. PC is obtained experimentally. We further propose a procedure for the improved derivation of GIC from the experimental results: According to fracture mechanics theory, the GIC vs. load function, Eq. (1), can be
presented as follows: 1 P2C dC ¼ GIC . da 2B Thus, plotting P2C =2B for different crack lengths against (dC/da)1 results in a linear plot that should intersect the origin, provided the joint’s behaviour is indeed linearelastic. The slope of this line represents the value of GIC. The graph of P2C =2B values obtained experimentally vs. (dC/da)1 calculated with Penado’s model, Eq. (3), for a DCB specimen made of aluminum/PG7770 is shown in Fig. 7(a). The same experimental results were used to obtain GIC values in accordance with the ASTM-D-3433 method, the result being shown in Fig. 7(b). It is clearly seen that the resulting line in Fig. 7(a) is linear and intersects the origin, which means that the behaviour of the system is indeed linear-elastic. The slope of this line is the GIC, and its value is also shown in Fig. 7(a). The average value of GIC as obtained using the ASTM standard, Fig. 7(b), is significantly lower than the one calculated with Eq. (3). Two other examples, each for a different system, are given in Figs. 7(c)–(f). The calculated GIC by the current method and by the ASTM standard of 7 additional adhesive joints are shown in Table 1. All the data show that GIC calculation based on ASTM standard underestimates the fracture energy, with much larger scatter. 4. Discussion
Fig. 6. The derivative of compliance vs. crack length for a DCB specimen using different models for a tough adhesive (a), and a ductile adhesive (b).
The existing standard [11] for measuring the fracture energy, GIC, of adhesive joints is based on linear elastic fracture mechanics and implicitly assumes that the adhesive layer’s parameters are negligible when calculating the compliance of the DCB specimen. The validity of this assumption was examined in the present investigation, and it was found to conflict with the experimental results obtained, specifically for joints that have a relatively thick adhesive layer. The compliance calibration method, which is frequently used, has two major disadvantages; the need for a large number of specimens and the requirement to reassess the compliance every time even a minor change is made in the specimens. Furthermore, the empirical results obtained include both the adherend’s and the adhesive’s contributions to the overall compliance, and it is impossible to distinguish between the two. The methods suggested in the recent round-robin [17] seem to overcome some of the problems existing in the ASTM standard, but do not resolve the need for an experimental evaluation of the compliance function. A method for calculating GIC is proposed in the present investigation. This method is based on combining the experimentally obtained measurements of the critical load for crack initiation with an analytical model to calculate the compliance of the specimens more accurately. It was found that the values obtained for GIC in our specimens were always higher than the GIC values obtained using the
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Fig. 7. GIC calculated in accordance with the proposed technique compared with that determined with the ASTM method for the following systems: aluminum/PG7770 DCB specimen—(a) and (b), aluminum/FM-73 DCB—(c) and (d), and aluminum/U-3150 DCB—(e) and (f).
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Table 1 GIC values of several adhesive joints as calculated using the suggested method and by ASTM D-3433 Adhesive used
GIC (suggested method) (J/m2)
GIC (ASTM D-3433) (J/m2)
Helmipur 700T/LF500 Hysol 4183/3561 E-433 Uralite 3125 Epon 828/V140 Fusor 305 EN-2
980770 682730 615740 260720 206710 188713 181720
8877179 457771 4127158 178746 195726 178770 125740
ASTM standard, due to the adhesive parameters in the latter being neglected. In fact, this result could be predicted from Figs. 6(a) and (b): the measured compliance and its derivative were always higher than those resulting from beam theory as used by the ASTM standard. Since GIC is proportional to the derivative of the compliance function, the resulting GIC will also be higher. It is evident that the ASTM method may be inadequate for measuring the fracture energy of adhesive joints with a thick and relatively compliant bond layer. Describing the DCB specimen as a beam on elastic foundation was found to give an accurate estimate for the compliance of DCB specimens. Finally, in our method the fracture energy was found to be independent of the crack length. It is noted that the elastic properties of the adhesives used are required for our calculations. Furthermore, our method applies only to adhesive joints that are within the LEFM framework. The possibilities to incorporate this method for non-linear adhesives or adhesives with R-curve behavior need further examination. 5. Summary The common practice for measuring GIC of adhesive joints, as described by the ASTM standard, has several
disadvantages. The typical results show a large diversity and can, moreover, lead to different values of GIC for different crack lengths. Such results are liable to preclude the use of GIC as a failure criterion and thus give rise to the use of the empirical compliance calibration technique, which is not very effective when testing many parameters of the adhesive joint. The approach used in this work introduces the use of a known analytical model for the improved calculation of the system’s compliance by including the adhesive’s parameters. References [1] Penado E. J Compos Mat 1993;27(4):383–407. [2] Liu Z, Gibson RF, Newaz GM. In: Proceedings of 31st International SAMPE technical conference, Society for the Advancement of Material and Process Engineering, Chicago, IL, October 1999. [3] Williams JG. Int J Fracture 1988;36:101–19. [4] Krenk S. Eng Fracture Mech 1992;43(4):549–59. [5] Fernlund G, Spelt JK. Composites Sci Tech 1994;50:441–9. [6] Kinloch AJ, Little MSG, Watts JF. Acta Mater 2000;48:4543–53. [7] Blackman BRK, Kinloch AJ, Taylor AC, Wang Y. J Mater Sci 2000;35(8):1867–84. [8] Curley AJ, Hadavinia H, Kinloch AJ, Taylor AC. Int J Fracture 2000;103:41–69. [9] Jacobsen TK, Sorensen BF. Composites: Part A 2001;32:1–11. [10] Trantina C. J Compos Mater 1972;6:371. [11] ‘‘Fracture strength in cleavage of adhesives in bonded metal joints’’, ASTM-D3433-99, Annual Book of ASTM Standards (American Society for Testing and Materials, Easton, MD). [12] Liechti KM, Freda T. J Adhesion 1989;28:145–69. [13] Park BY, Kim SC, Jung B. Polymers Adv Technol 1996;8:371–7. [14] Rakestraw RD, Vrana MA, Dillard DA, Ward TC, Dillard JG. A fracture mechanics methodology for the testing and prediction of adhesive bond durability, AD-Vol 43, Durability and Damage Tolerance, ASME 1994. [15] Butkus LM. Environmental Durability of Adhesively Bonded Joints, Ph.D. Thesis, Georgia Institute of Technology, 1997. [16] Dalmas D, Laksimi L. Appl Compos Mater 1999;6:327–40. [17] Blackman BRK, Kinloch AJ, Paraschi M, Teo WS. Measuring the Mode I adhesive fracture energy, GIC, of structural adhesive joints: the results of an international round-robin. Int J Adhesion Adhesives 2003;23:293–305.